In addition, we include in Fig. REF  a second estimate that can be derived in the magnetic picture and which we will refer to as the maximal estimate [1]}.
This estimate is based on the evolution equation for the energy density of the electromagnetic field, which contains an additional source term in the presence of the axion,
\(\left(\frac{1}{a}\partial _\tau + 4H\right)\frac{E^2 + B^2}{2a^4} = 2H\,\frac{\xi _{\rm eff}\,EB}{a^4} \,.\) 
